1. ## Number Pattern

Consider the pattern.

$\displaystyle 5^3=125$

$\displaystyle 5^4=625$

$\displaystyle 5^5=3125$

$\displaystyle 5^6=15625$

Use this pattern to write out the last 3 digits of $\displaystyle 5^{15}$.

How can I work this out?

Consider the pattern.

$\displaystyle 5^3=125$

$\displaystyle 5^4=625$

$\displaystyle 5^5=3125$

$\displaystyle 5^6=15625$

Use this pattern to write out the last digits of $\displaystyle 5^{15}$.

How can I work this out?
1. You do not specify how many digits constitute "the last" digits of $\displaystyle 5^{15}$.

2. A suitable hypothesis might be that the last two digits of $\displaystyle 5^n$ for $\displaystyle n\ge 2$ are $\displaystyle 25$.

3. Suppose this true for $\displaystyle k$ that is $\displaystyle 5^k=100 \times M+25$ for some integer $\displaystyle M$, now $\displaystyle 5^{k+1}= ...$

NOw can you take it from there?

CB

3. Sorry, it was the last three digits. This was easier than I thought! How silly of me!!

Seeing as odd indices always result in ending digits of 125, and even indices result in 625, the answer to this problem would be 125. Am I right?